Implicit Differentiation

Date: 11/1/96 at 19:12:28
From: brenchi
Subject: Implicit Differentiation

Given that (x^p)*(y^q) = (x+y)^(p+q), prove that: dy/dx = y/x.

Thanks a lot!

From Billy                   

Date: 11/2/96 at 16:39:1
From: Doctor Anthony
Subject: Re: Implicit Differentiation

Starting with    (x^p)*(y^q) = (x+y)^(p+q),       

  take logs of both sides:

           p.ln(x) + q.ln(y) = (p+q)ln(x+y)   

  differentiate implicitly:

           p/x + (q/y).dy/dx = (p+q)/(x+y)[(1 + dy/dx)]
   
  collect terms in dy/dx:

    dy/dx[q/y - (p+q)/(x+y)] = (p+q)/(x+y) - p/x

 dy/dx[(qx+qy-py-qy)/y(x+y)] = (px+qx-px-py)/x(x+y)

             dy/dx (qx-py)/y = (qx-py)/x

                       dy/dx = y/x

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/